Puzzle for October 10, 2022 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
Scratchpad
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Hint #1
In eq.6, replace C with D + F (from eq.4): D + F + F = D + E which becomes D + 2×F = D + E Subtract D from each side of the equation above: D + 2×F - D = D + E - D which makes 2×F = E
Hint #2
In eq.3, replace E with 2×F: B = 2×F + F which makes B = 3×F
Hint #3
In eq.5, substitute 3×F for B, and 2×F for E: D = 3×F + 2×F which makes D = 5×F
Hint #4
Substitute 5×F for D, and 2×F for E in eq.6: C + F = 5×F + 2×F which becomes C + F = 7×F Subtract F from each side of the equation above: C + F - F = 7×F - F which becomes C = 6×F
Hint #5
Substitute 3×F for B, and 6×F for C in eq.2: A = 3×F + 6×F which makes A = 9×F
Solution
Substitute 9×F for A, 3×F for B, 6×F for C, 5×F for D, and 2×F for E in eq.1: 9×F + 3×F + 6×F + 5×F + 2×F + F = 26 which simplifies to 26×F = 26 Divide both sides of the above equation by 26: 26×F ÷ 26 = 26 ÷ 26 which means F = 1 making A = 9×F = 9 × 1 = 9 B = 3×F = 3 × 1 = 3 C = 6×F = 6 × 1 = 6 D = 5×F = 5 × 1 = 5 E = 2×F = 2 × 1 = 2 and ABCDEF = 936521